#9065936

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#40944	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4112 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2)(3,9)(4,7)(5,13)(6,8)(10,14)(11,15)(12,16)(17,18,23,27)(19,24,28,30)(20,22,26,32)(21,25,31,29)(33,34)(35,41)(36,40)(37,38)(39,42)(43,46)(44,45)(47,48);(1,3,6,4)(2,7,8,9)(5,11,14,12)(10,15,13,16)(17,19)(18,24)(20,29)(21,22)(23,28)(25,26)(27,30)(31,32)(33,35,39,36)(34,40,42,41)(37,43,47,45)(38,44,48,46);(2,8)(7,9)(10,13)(15,16)(19,28)(21,31)(24,30)(25,29)(33,35,39,36)(34,41,42,40)(37,45,47,43)(38,44,48,46);(1,4,6,3)(2,9,8,7)(5,11,14,12)(10,15,13,16)(17,20)(18,22)(19,29)(21,24)(23,26)(25,28)(27,32)(30,31)(33,36,39,35)(34,41,42,40)(37,43,47,45)(38,44,48,46);(1,2)(3,9)(4,7)(5,10)(6,8)(11,16)(12,15)(13,14)(17,21)(18,25)(19,22)(20,30)(23,31)(24,26)(27,29)(28,32)(33,37)(34,38)(35,43)(36,45)(39,47)(40,44)(41,46)(42,48);(1,5)(2,10)(3,11)(4,12)(6,14)(7,15)(8,13)(9,16)(17,22)(18,26)(19,21)(20,27)(23,32)(24,25)(28,31)(29,30)(33,38)(34,37)(35,44)(36,46)(39,48)(40,43)(41,45)(42,47)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,d^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8973132);